On structural properties of trees with minimal atom-bond connectivity index

The {\em atom-bond connectivity (ABC) index} is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal $ABC$ index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more than four $B_4$-branches.